The Stacks project

Lemma 48.27.5. Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $d$. The module $\omega _ X$ of Lemma 48.27.1 has the following properties

  1. $\omega _ X$ is a dualizing module on $X$ (Section 48.22),

  2. $\omega _ X$ is a coherent Cohen-Macaulay module whose support is $X$,

  3. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X[d]) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, K), k)$ compatible with shifts and distinguished triangles for $K \in D_\mathit{QCoh}(X)$,

  4. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^{d - i}(\mathcal{F}, \omega _ X) = \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$.

Proof. It is clear from Lemma 48.27.1 that $\omega _ X$ is a dualizing module (as it is the left most nonvanishing cohomology sheaf of a dualizing complex). We have $\omega _ X^\bullet = \omega _ X[d]$ and $\omega _ X$ is Cohen-Macaulay as $X$ is Cohen-Macualay, see Lemma 48.23.1. The other statements follow from this combined with the corresponding statements of Lemma 48.27.1. $\square$

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